Garay, Pedro V.

Golden, Bill B.

Peterson, Jeffrey M.

Smith, Craig M.

English

Research Papers in Economics

1 
 
 
 
 
Disaggregated Spatial Modeling of Irrigated Land and 
Water Use 
 
 
Pedro V. Garay
1, 
Jeffrey M. Peterson
2, 
Craig Smith
3, 
Bill Golden
4 
 
 
                                                         May 4, 2010 
 
 
 
 
 
 
 
 
 
 
Selected Paper prepared for presentation at the Agricultural & Applied Economics 
Association Annual Meeting, July 23 ‐25, 2010. 
 
 
 
 
 
 
 
 
                                                 
1 Graduate Research & Teaching Assistant, Department of Agricultural Economics, Kansas State University  
2 Associate Professor, Department of Agricultural Economics, Kansas State University  
3 Graduate Research Assistant, Department of Agricultural Economics, Kansas State University 
4 Research Assistant Professor, Department of Agricultural Economics, Kansas State University 2 
 
Introduction 
Water scarcity is already a critical issue across much of the world, and it will become an 
even greater concern in the semi-arid region of the eight U.S. states overlying the Ogallala 
aquifer. The Ogallala aquifer is one of the largest water resources in the world, underlying 111.6 
million acres including portions of Colorado, Kansas, Nebraska, New Mexico, Oklahoma, South 
Dakota, Texas and Wyoming (U.S. Geological Survey, 2006). This area represents one of the 
most important agricultural regions in the nation.  
The Ogallala aquifer contributes to the development of agricultural industries including 
irrigated crops, cattle feeding, and meat processing. After reliable technology became available 
to withdraw groundwater for irrigation in the mid Twentieth Century, the acreage of irrigated 
crops increased dramatically. As the Ogallala aquifer is the principal source of water for 
irrigation in the High Plains region, a natural consequence of a high demand of this resource for 
irrigation is a decline of water level availability. The area-weighted average water-level declined 
14.0 feet from 1950 to 2007 (McGuire, 2009). Historically, the portion of the aquifer in Kansas 
has experienced one of the highest levels of depletion. For instance, for the period between 1950 
and 2007 the water level change in the portion of the aquifer in Kansas declined 22.7 feet.  
While there is an extensive literature developing economic simulation models of irrigated 
land and water use (see Brouwer and Hofkes (2008) and Harou et al. (2009) for recent reviews), 
little attention has been paid to the appropriate level of aggregation for these models.  In most 
cases, researchers model the representative land unit in a relatively large region.  This approach 
ignores any spatial variability within the region, which masks distributional issues that may be 
important for policy and, if many if the model relationships are highly nonlinear, may increase 3 
 
the model’s prediction error.  In particular, even if one is only interested in aggregate measures at 
the regional scale, a more accurate prediction may be obtained by constructing many versions of 
the model – with each version calibrated to a distinct spatial unit—and then aggregating the 
results.  
The main objective of this paper is to determinate the level of data aggregation that can 
give us the most accurate prediction of observed aggregates. To do so, we analyze the results of a 
Positive Mathematical Programming (PMP) model calibrated to data at varying levels of 
aggregation. In the first step of this approach, crop choices and water use at three level of data 
aggregation (county, crop report district and state aggregation) are simulated by a Positive 
Mathematical Programming (PMP) model (Howitt, 1995; Clark, 2008). A separate version of 
this model calibrates the observed crop allocation and water use in the base year (average from 
2000 to 2008) to the three levels of data aggregation for comparison. This allows us to compare 
the approaches of (a) running a model with county-level data and aggregating the results to the 
state level, (b) running a model with data aggregated to multi-county crop reporting districts and 
then aggregating the results on further step to the state level and (c) running the models on state-
level aggregate data.  
Comparing the results and analyzing their prediction errors for the observed years 
following the calibration year (2000-08), allows us to test for aggregation bias. The measures 
that this paper uses to evaluate which of the simulation models fit better with the actual situation 
are the root-mean-square simulation error and the mean simulation error. The preliminary results 
suggest that the more disaggregated the data used in the calibration and simulation processes, the 
model fits better with the reality.  4 
 
The rest of this paper is organized as follows. First, the models and methods are 
discussed. Second, the database that is used to calibrate and simulate the models is presented. 
Next, the results of the simulation at different levels of data aggregation are presented. The final 
section of the paper summarizes and concludes. 
 
Methods and Procedures 
The specific approach considered for this paper is The Positive Mathematical 
Programming (PMP). This method was initially used on policy model at different levels of 
production such as sectoral, regional and farm level. A peculiar aspect of this method is that, in 
spite several papers had used this method; it was not published until 1995 by Richard E. Howitt. 
In this article Howitt explains the characteristics and advantages of the PMP approach. An 
advantage of the PMP model is that, once calibrated, its solutions reproduce the observed 
patterns within a small tolerance, without adding ‘flexibility’ constraints in the model to 
artificially limit crop acreages (Howitt, 1995). This characteristic has made PMP a widely-used 
approach for policy modeling. 
Our study region is a 31-county area overlying the Ogallala aquifer in western Kansas.  
We refer to this as the “region” level of aggregation in what follows.  This region encompasses 
approximately the western third of Kansas. Following the boundaries of the USDA-NASS crop 
reporting districts, this region can be subdivided into three multicounty areas (including the 
northwest, west central, and southwestern Kansas crop reporting districts).  These boundaries 
yield the intermediate “district” level of aggregation.  
This paper applies the PMP model developed by Clark (2008) for irrigated agriculture in 5 
 
western Kansas.  Our analysis proceeds in three major steps.  First, the PMP model is calibrated 
to observed crop allocation and water use in the base period (a nine-year average from 2000 to 
2008) at each of the three levels of data aggregation. Second, the calibrated models are simulated 
over the data period, allowing prices to fluctuate yearly. A difference between this paper and 
Clark 2008 is that this paper uses the base period average price for calibration purpose, while for 
the simulation process; the yearly prices from the Kansas State University Extension budgets are 
used. Therefore, the prices for the simulation process are not constant. Third and finally, the 
results of the county- and district-level models are aggregated up to the region level so that their 
accuracies in predicting aggregate measures can be compared. 
The model from Clark (2008) predicts land and water allocation for eight crops (irrigated 
wheat, nonirrigated wheat, irrigated corn, nonirrigated corn, irrigated sorghum, nonirrigated 
sorghum, irrigated soybean, and irrigated alfalfa).  
The famer land and water allocation problem is represented as follows
5: 
   ∑                       ,   ;   ,   ,            ( 1 )  
s.t. ∑          
where    b = size of the farm, 
         = production function for crop i,  
   = output price for crop i, 
   = land area planted to crop i, 
                                                 
5 The complete version of the model can be finding in Clark (2008). 6 
 
    = water use per crop i,  
      ,   ;   ,   ,    = cost function for crop i, and 
   ,   ,   = parameters of the cost function. 
The production function that is used in this paper was developed by Martin, Watts, and 
Gilley (1984). The cost function is linear in both land allocations and water use, and it was 
specified by Clark (2008). The cost parameters are obtained from the calibration process and 
represent part of the necessary input for the simulation part. The model that is simulated for a 
period of t = 1… 9  years can be represented as follow: 
max∑   ,          ,    ,           , ,   , ,     ,     ,           ,    ,               (2) 
 s.t.  ∑   ,         
                 ∑   ,           
                         ,                           
where Q = set of indices of irrigated crops, 
  mt = maximum pumping given aquifer conditions in year t 
     = legally authorized irrigated acreage, 
              = extra cost of pumping per year, 
    ,  = output price per crop and per year. 7 
 
Several measures can be calculated to assess a model’s prediction accuracy. Willmott et 
al. (1985) mentioned different measures that can be estimated to determinate both model 
accuracy and precision. In the set of different measures they emphasize the approaches based in 
the difference between the elements and values of a model predicted and observed. These 
measures include the root-mean-square error (RMSE), the systematic root-mean-square error 
(RMSEs), the unsystematic root-mean-square error (RMSEu), and the index of agreement (d2). 
They also present additional indices such as the mean absolute error (MAE) and a modified 
version of the index of agreement (d1). They also argued that bootstrapping can be used to 
evaluate both the confidence and significance associated with each of the difference indices. 
They showed that if the difference measures mentioned above are used in combination with the 
correct statistics and data-display graphics the evaluation of the performance of models can be 
accomplished. 
Let Y denote some endogenous variable of interest that is simulated by the model. In this 
analysis we calculate and report the root-mean-square simulation error using the following 
formula (Pindyck and Rubinfeld, 1998): 
             
 
 ∑     
      
     
            ( 3 )  
where     
  = simulated value of     
     
  = actual value 
  T = number of periods in the simulation  
  The root-mean-square-simulation error (RMSE) measures the differences between the 
values predicted by a simulation model and the actual value of the studied variable. The RMSE 8 
 
is measured in the same units as the data, and is representative of the size of a typical error. 
Additionally, the root-mean-square percent simulation error (RMSPE) is calculated 
following: 
                     
 
 ∑  
   
    
 
   
   
 
                                                    (4) 
  The root-mean-square percent error provides the same properties as the root-mean-
squares error, but expressed as percents. 
Data 
This paper uses county- level data from different sources. Crops acres and yields from 
2002 to 2008, for the thirty-one counties located in the northwest, west-central and southwest of 
Kansas, are from the National Agriculture Statistics Service (NASS). The base period for the 
calibration and simulation models is the average of this nine-year period 2000-08.  
The crop prices and the cost variables are from Kansas State University Extension 
budgets. The crops price are expected prices and the cost variables, which are included in the 
model are irrigation costs, variable expenses, fertilizer and seed costs, and harvest and hauling 
cost. The aquifer level data are from the Kansas Geological Survey. The variables which are 
considered from the later source are lift, saturated thickness, specific yield, land area above 
aquifer, hydraulic conductivity, annual recharge rate, and depth. The number of wells is from the 
Water Information Management and Analysis System (WIMAS).  
Another important group of data is the agronomic data, which are obtained from several 9 
 
sources. Agronomic variables required for the model include actual irrigation water use, 
irrigation required for fully watered yield, precipitation level, gross and net irrigation 
requirement, fully watered yield, and dryland yield. These data were obtained from the Water 
Information Management and Analysis System, NOAA National climatic Data Center, Kansas 
Weather Data Library, The National Engineering Handbook, Stone et. al; and O’Brien et. al. 
Results 
  The models calibrate and simulate crop allocation and water use for eight different crops. 
As the main objective of this paper is to evaluate which of the three simulation models fits better 
with the actual value of the variables, irrigated corn acreage is selected for comparison purposes. 
The following graph shows the simulation results obtained from the three models and the actual 
value of the irrigated corn from 2000 to 20008. 
 
                                  Figure 1 
0
200,000
400,000
600,000
800,000
1,000,000
1,200,000
1,400,000
1,600,000
1,800,000
2,000,000
2000 2001 2002 2003 2004 2005 2006 2007 2008
Observed and predicted irrigated corn acres, 
western Kansas
Irr Corn Acres actual Irr Corn State Sim Irr Corn Dist Sim Irr Corn Count Sim10 
 
  The blue line reflects the actual value of irrigated corn acreage corresponding to the 31-
county region obtained from the National Agriculture Statistics Service (NASS) for the period of 
2000 to 2008.  The red line represents the simulated results of irrigated corn using the most 
aggregated data. This model is called state level simulation.  The green line reflects the result 
obtained from the model that used the district level aggregation.   The value was obtained adding 
the irrigate corn acreages from each district. This model represents the first level of data 
aggregation and it is called district simulation model. Finally, the purple line represent the results 
of the third model called county level simulation. The value of this line is the summation of 
irrigate corn acreages obtained in the simulation for each of the thirty-one counties.  
  To make a more precise evaluation about the fit of three models, the root-mean-square 
simulation errors and root-mean-square-percent simulation error are presented in the following 
table. 
Table 1. Model Prediction Errors for Region-Level Irrigated Corn Acreage, 2000-08 
 
Model 
 
RMSE (acres) 
 
RMSPE (percent) 
 
State Level Aggregation 
294,911 
 
28.37 
 
District Level Aggregation 
264,777 
 
25.83 
 
County Level  
254,129 
 
24.28 
                                
  The root-mean-square-simulation-error (RMSE) is a measure of the differences between 
values predicted by a simulation model and the actually values observed from the variables 
modeled. It is a measure of precision. The results of RMSE presented in the above table show 11 
 
that the model that used the less level of data aggregation fits better with the actual value of the 
variable than the others two models. In the same sense, the district level aggregation model fits 
better than the state level model. These results are consistent with the second measure of 
precision represented by the root-mean-square percent error. Both the RMSE and the RMSPE are 
consistent with the expected results under the argument that the more aggregated the data, the 
probability of getting sample errors are higher.  
Conclusion 
The main objective of this paper was to determinate the level of data aggregation that can 
give us a better fit of a simulation model to actual value of the variable studied. This paper 
analyzed the data aggregation issue using a Positive Mathematical Programming (PMP) model 
constructed by Clark (2008). The data to calibrate and simulate the models were obtained from 
NASS, Kansas State Research and Extension, the Kansas Geological Survey, the Water 
Information Management and Analysis System, and Kansas Weather Data Library.  
The models were calibrated with data aggregated to the county, crop reporting district, 
and region level for comparison. This allowed us to compare the approaches of running 
disaggregated models and then aggregating the results, and running the models on aggregate 
data. To analyze the results, the root-mean-square simulation error for each model was 
calculated. For simplicity, just the irrigate corn acreages was chosen as variable of comparison.  
The results of the tests suggest that model that is simulated with most disaggregated data 
gives us the smallest value of the RMS simulation error, and the smallest percentage error, 
implying that for this specific variable (Irrigate corn acreages) the data aggregation can generate 12 
 
a big difference between the values predicted by the model and the values actually observed. 
Consistent with this result, a graphical analysis shows that the results obtained from models run 
with disaggregated data follow more closely the actual value pattern of the selected crop.  
This paper can be considered as initial study for future researches which can provide 
information about the ideal level of aggregation for different types of data such as dryland crops 
and water use.   
  13 
 
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Ogallala Aquifer”, M.S. Thesis, Department of Agricultural Economics, Kansas State 
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Positive Mathematical Programming.&quot;

Statistics for the Evaluation and Comparison of Models”,

Survey

Disaggregated Spatial Modeling of Irrigated Land and Water Use

Disaggregated Spatial Modeling of Irrigated Land and Water Use

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